Endpoint regularity of general Fourier integral operators

Let \(n\ge 1,0<\rho <1, \max \{\rho ,1-\rho \}\le \delta \le 1\) and

$$\begin{aligned} m_1=\rho -n+(n-1)\min \{\frac{1}{2},\rho \}+\frac{1-\delta }{2}. \end{aligned}$$

If the amplitude a belongs to the Hörmander class \(S^{m_1}_{\rho ,\delta }\) and \(\phi \in \Phi ^{2}\) satisfies the strong non-degeneracy condition, then we prove that the following Fourier integral operator \(T_{\phi ,a}\) defined by

$$\begin{aligned} T_{\phi ,a}f(x)=\int _{{\mathbb {R}}^{n}}e^{i\phi (x,\xi )}a(x,\xi ){\widehat{f}}(\xi )d\xi , \end{aligned}$$

is bounded from the local Hardy space \(h^1({\mathbb {R}}^n)\) to \(L^1({\mathbb {R}}^n)\). As a corollary, we can also obtain the corresponding \(L^p({\mathbb {R}}^n)\)-boundedness when \(1<p<2\). These theorems are rigorous improvements on the recent works of Staubach and his collaborators. When \(0\le \rho \le 1,\delta \le \max \{\rho ,1-\rho \}\), by using some similar techniques in this note, we can get the corresponding theorems which coincide with the known results.